Cylinder Pressure in a Spark-Ignition Engine - Harvard Computer ...

J. Undergrad. Sci. 3: 141-145 (Fall 1996)
Engineering Sciences
Cylinder Pressure in a Spark-Ignition Engine: A
Computational Model
PAULINA S. KUO
The project described in this article attempts to accurately
predict the gas pressure changes within the cylinder
of a spark-ignition engine using thermodynamic
principles. The model takes into account the intake, compression,
combustion, expansion and exhaust processes
that occur in the cylinder. Comparisons with
actual pressure data show the model to have a high
degree of accuracy. The model is further evaluated on
its ability to predict the angle of spark firing and burn
duration.
Introduction
Gas pressure in the cylinder of an engine varies throughout
the Otto four-stroke engine cycle (see ref. 1 for an elaboration
of this process). Work is done on the gases by the
piston during compression and the gases produce energy
through the combustion process. These changes in energy
combined with changes in the volume of the cylinder lead to
fluctuations in gas pressure. The ability to accurately predict
the pressure allows for better understanding of the processes
taking place in the cylinder such as the interactions
between the gases, oil film, piston and liner. In the first phase
of this project, a computer model was developed to predict
pressure based on initial conditions and engine geometries.
A problem encountered in gathering data was the fact
that the spark timings were not accurately known due to
variations in engine speed. The exact moments of spark
firing and cessation of combustion were unknown. Spark
firing was estimated to be at 25 ° to 5 ° before top center (BTC)
and the burn duration is approximately 60 ° to 80 ° . The second
phase involved determining the most precise values for
spark firing and burn duration by comparing the model to
actual data.
The Model
The model, which was programmed in FORTRAN, predicts
the cylinder pressure throughout the intake, compression,
combustion, expansion and exhaust processes that
make up the engine operating cycle. Pressure was modeled
as a function of the angle of the crank (see ref. 1),
which ran for 720 degrees per cycle or two revolutions because
the crank completed two rotations per cycle. The valve
and spark timings, engine geometry, engine speed and inlet
pressure were entered into the model.
The individual processes of the engine cycle, intake,
compression, combustion, expansion and blowdown/exhaust,
are discussed below in order of occurrence.
Intake. Intake occurs between exhaust valve closing (EVC)
and the start of compression. The intake valve opens before
the exhaust valve closes, so there is a period of overlap during
which both valves are open. During this period, the model used
an s-curve to describe the gradual transition between exhaust
pressure and intake or inlet pressure. The start of compression,
which marks the end of the intake process, does not necessarily
occur at the same time as the close of the intake valve
(IVC). The intake valve closes after bottom center (BC) while
the volume in the cylinder is decreasing. The engine speed
determines the point at which the fuel/air mixture stops flowing
into the cylinder. At lower revolutions per minute (rpm), the start
of compression is closer to IVC; at higher speeds, it is closer to
BC. An s-curve approximates the angle of the start of compression
as a function of engine speed.
The volume of the cylinder during intake increases as
the piston descends, thereby drawing in the fuel mixture.
There is little resistance to gas flow into the cylinder, which
causes the pressure in the cylinder to remain relatively constant
and equal to the inlet pressure.
Compression. Both the intake and exhaust valves are
closed during the compression stroke so that the gases can
neither enter nor exit the cylinder. The piston is moving upward,
so cylinder volume decreases. Pressure increases
as the gas in the cylinder is compressed. Because of the
high speed of the piston, the duration of compression is short
and negligible heat is lost to the walls of the cylinder. Relatively
little energy is dissipated due to internal friction of the
gas. Overall, there is little change in entropy during compression,
and the gas behavior can be described by the
equation: 2 (1)
This equation allowed for calculation of the cylinder pressure
at any crank angle during compression based on the
knowledge of initial pressure and volume, P 0
and V 0
, which
determine the constant. The volume of the cylinder is a direct
function of crank angle, cylinder geometries, crank radius
and connecting rod length (see ref. 1). The ratio of the
specific heat of the fuel at constant pressure to the specific
heat at constant volume is γ; its value varies from compression
to combustion to expansion. During compression, γ is
approximately equal to 1.3. 3
Combustion. The combustion process was described by
the McCuiston, Lavoie and Kauffman (MLK) model. 4 The
mass-burn fraction, χ b
, was modeled as a function of pressures
and volumes:
where
P = pressure corresponding to burn fraction
V = volume corresponding to burn fraction
P 0
= pressure at start of combustion
V 0
= volume at start of combustion
P f
= pressure at end of combustion
V f
= volume at end of combustion
n = polytropic constant
(2)
PAULINA S. KUO is a recent graduate of Thomas Jefferson High School for Science and Technology. She completed the research described in this article under the supervision of
Professor John Heywood at the MIT Sloan Automotive Laboratory. Ms. Kuo was honored as a Westinghouse Science Talent Search finalist for this work, and she was a member of the
1996 US Physics Team; she aspires to a career as an engineer or a professor.

142 Journal of Undergraduate Sciences
Engineering Sci.
The polytropic constant, n, is an empirically determined constant
about equal to γ which has a value close to 1.25 during
combustion. 3
The fraction of fuel burned varies with an s-curve which
runs from 0% burned at the start of combustion to 100%
burned at the end of combustion. The Weibe function is an
s-curve that is used to describe the burn fraction: 1 (3)
θ 0
is the angle at which combustion begins; it is about equal
to the angle of spark firing. The angle of the duration of combustion
is ∆θ. The constants a and m are determined experimentally.
Real burn fraction curves have been fitted by
the Weibe function with a=5 and m=2.
The terms in equation (2) can be rearranged to give an
expression for pressure in terms of the crank angle and the
conditions at the onset and end of combustion. The final
conditions can be represented as functions of the initial conditions
by using the fact that the gases in the cylinder act
almost ideal and that negligible energy is lost or gained by
the system. Since the combustion process occurs almost
symmetrically about top center (TC), the volume of the cylinder
at the spark is approximately equal to the volume at
the end of combustion. Because the initial and final volumes
are about equal, little net work is done on the piston and the
change in temperature between the spark and end of combustion
is due to the burning of the fuel.
According to the ideal gas law, the gas pressure at the
end of combustion, P f
, is a function of the gas constant of
the fuel, mass of the gases, temperature, and volume. The
gas constant of the fuel is equal to the universal gas constant
divided by the molar mass of the fuel. The total mass
of the gas mixture was calculated at IVC, just when the cylinder
is sealed, by using the ideal gas law. Temperature at
the end of combustion was estimated using the assumption
that nearly all of the chemical energy of the fuel caused a
temperature change of the gases. This idea is expressed in
the following equation: 5 (4)
gases drives the piston down, does work by turning the crankshaft
and, in turn, provides power to the car. During expansion,
the heat transferred to the cylinder liner is small compared
to the work done. Energy lost to internal friction of the
gas is also minimal. The gas behavior was described by
equation (1) which is also used to model the compression
process. The conditions at the end of combustion determined
the constant term. During expansion, γ has a value of about
1.48. 3
Blowdown and Exhaust. Exhaust valve opening (EVO) occurs
before the crank reaches BC. At this point, the pressure
in the cylinder is much greater than the exhaust system
pressure. The higher pressure in the cylinder helps push
the burnt gases out of the cylinder. The process by which
the pressure aids in the expulsion of burnt gases is called
blowdown. The flow of the gases can be described by a
model of gas flow through an orifice where the valve acts
like the flow restriction. 1 This model depends on the velocity
of the gas. When the gas velocity at the smallest portion of
the opening, the throat, is equal to the speed of sound, the
flow is said to be choked. The flow rate under choked flow is
described by:
where
(5)
.
m real
= real mass flow rate
C D
= discharge coefficient
A T
= cross-sectional area of throat
p T
= pressure at throat
p 0
= pressure in the cylinder
T 0
= temperature in the cylinder
R = characteristic gas constant of the exiting gases
γ = ratio of specific heats of the exiting gases
Subsonic flow is described by the equation:
(6)
where
m total
= total mass of gases in the cylinder
c v
= specific heat of gas mixture at constant vol.
T f
= temperature at the end of combustion
T spark
= temperature at the start of combustion
C = coefficient for unburned fuel
m fuel
= mass of fuel in the cylinder
Q HV
= heating value of the fuel
The temperature at the spark was calculated by applying
the ideal gas law to the conditions at the end of compression.
The constant C took into account the fact that not all of
the fuel was burned and thus not all of the chemical energy
available was changed into thermal energy; C is approximately
0.95. The mass of the fuel was calculated by using
the air-to-fuel ratio and noting that the total mass is equal to
the mass of air plus the mass of fuel. The heating value is a
constant dependent upon the type of fuel used. For gasoline,
the heating value is about 44 megajoules per kilogram.
Expansion. The end of combustion occurs slightly after TC
at which point expansion begins. The pressure of the burned
The discharge coefficient is an experimentally determined constant
which relates the effective throat area to the actual throat
area. It is roughly equal to 0.7. Secondary flow near the throat
cause the main flow to pass through a smaller area than the
actual cross-sectional area of the throat. The pressure at
.
the
throat is equal to that of the gases in the exhaust system. m, p 0
and T 0
are variable but can be related by the ideal gas law:
and the isentropic process gas model: 2 (8)
These models are valid because there is negligible entropy
change in the system and the gases in the cylinder do not
deviate far from ideal behavior.
The resultant equation
.
is a differential equation where
the mass flow rate, m, is dependent on the mass of the gases
in the cylinder, m. The value of m is computed numerically
by using the three-step Runge-Kutta algorithm and then used
to find the gas pressure.
(7)

143 Journal of Undergraduate Sciences
Engineering Sci.
The flow of the gases depends on the area of the opening
of the exhaust valve. This area changes as the valve is
pushed open and closed. It increases quickly to a maximum,
reaches a plateau and falls off again as the valve closes.
The maximum area is determined mainly by the shape of
the duct to the exhaust system. Pressure in the cylinder
settles down to the exhaust system pressure as the exhaust
valve remains open. The valve closes after TC after the next
engine cycle has begun.
The exhaust system pressure is a function of engine
speed. There are several barriers to the exiting gas in the
exhaust system, such as the catalytic converter and the
muffler, which resist the flow of the burnt gases and cause
the pressure in the exhaust system to increase roughly as
the square of the speed of the exiting gases. The speed of
the gases is directly proportional to the engine speed. Generally,
the exhaust pressure is between 1 atmosphere (atm)
and 1.5 atm.
Table 1. Model inputs. BTC stands for “before top center.”
Table 2. Analysis of 2000 rpm HL. ATC stands for “after top center.”
Model Evaluation
Data was collected from a Renault production engine
at different engine speeds and inlet pressures. The geometries
of the engine were entered into the model and the
model was then used to predict the pressure in the cylinder.
Specifications of the test engine are located in Appendix A.
The angle of spark firing and burn duration angle were not
explicitly known. Reasonable estimates for the timing were
made from prior experiences and by visual approximation.
The model was fairly accurate in its cylinder pressure
predictions. It was tested with engine pressure data at 2000
rpm full load (FL), 2000 rpm half load (HL), 4000 rpm FL,
and 4000 rpm HL. The first figure in Appendix B plots the
actual and model data at 4000 rpm FL. Of the four trials, the
model deviated the most at 4000 rpm FL. The differences of
all four trials between the model data and the actual data
are shown in the second figure of Appendix B. Differences
were greatest at higher rpm and greater load.
Under full load, the inlet pressure was about 1 atm and
0.5 atm at half load. The spark timing was changed from
model to model. Generally, an increase in engine speed was
accompanied by earlier spark firing and longer combustion
duration. An increase in inlet pressure meant that the spark
fired later, and the burn duration was shorter. The exact values
of the inlet pressure in kilopascals (kPa) and the spark
timing used in each model are given in Table 1.
The model results were compared to actual engine data.
Differences in P spark
, θ peak pressure
, P peak
and P EVO
were recorded
and analyzed. The accuracy of P spark
is a measure of the
performance of the compression model. Discrepancies in
θ peak pressure
and P peak
gauge the model of combustion while
P EVO
shows how well the expansion model described the
expansion process. Tables 2 through 5 show the performance
of the model under different engine conditions.
The compression model produced pressure values
slightly greater than the actual values. The percent error of
P spark
was around 10%. In an actual engine, some energy is
lost during the compression process. Because the energy
loss was not taken into account in the model, the pressure
predictions at the spark firing angle were larger than the
actual pressure values.
In some trials, the predicted angle of peak pressure in
the model was less than the actual angle. Because the encoder
on the engine that records the crank angle is sometimes
faulty at high engine speed, the actual pressure data
Table 3. Analysis of 2000 rpm FL. ATC stands for “after top center.”
Table 4. Analysis of 4000 rpm HL. ATC stands for “after top center.”
Table 5. Analysis of 4000 rpm FL. ATC stands for “after top center.”
may be slightly inaccurate. The peak pressure was usually
accurate but was at times lower than the actual value. These
variations were, in part, due to the uncertainties in the angle
spark firing and burn duration.
The cylinder pressure at EVO was the most variable.
The model incorporated a representation for blowdown,
which caused a visible drop off at EVO in the plots of the
model, but the actual data did not show such a large decrease.
The differences were more pronounced in the trials
at higher engine speeds. These discrepancies in pressure
during combustion and expansion were partially due to the
mechanics of the pressure transducer. The transducer does
not work well at high temperatures or at quickly changing
temperatures. Because of the rapidly changing conditions
in the cylinder, it may not have recorded all the changes in
pressure during blowdown.
Another reason for the differences is the inaccuracy of
the function that describes the area of the valve opening.

144 Journal of Undergraduate Sciences
Engineering Sci.
Table 6. Load inlet pressures.
Table 7. Optimal spark timings. BTC stands for “before top center.”
Without actually measuring the area of the valve opening in
the test engine, it is impossible to predict the changes in the
size of the opening. Valves can be opened by more than
one mechanism. The valves are opened at different rates,
depending on the mechanism. Hence, the time it takes for
the area to reach the maximum varies from engine to engine.
The curve that predicted the area of the valve opening
may have been incorrect for the test engine.
On the whole, the model predicted the pressure in the
cylinder fairly accurately. The shape of the curve was well
matched with the actual data. Pressure rose through compression,
peaked during combustion and fell during expansion
and blowdown. The standard deviation of the model
was 1.33 for 2000 HL, 2.40 for 2000 FL, 3.13 for 4000 HL
and 8.61 for 4000 FL. Considering that the pressure varies
from 50 kPa to 5000 kPa, the model was very accurate.
Spark and Burn Duration Determination
The exact angle of spark firing and duration of combustion
were not known. The engine cycles are completed extremely
quickly and the spark timing varies slightly each cycle
which made it difficult to accurately measure the spark timing.
By varying spark firing and burn duration inputs of the
model, the best fit curve was obtained that gave a prediction
of the spark timing.
Data Fitting. The error of the model is calculated by an sum
of the differences between the real and model pressure
values:
Data was sampled at integral crank angles. The error was
minimized to find optimal values for the angle of spark firing
and burn duration. Pressure data came from a Renault production
engine (see Appendix A for specifications). The pressure
measurements were taken at 2000 rpm under quarter
load, half load, three-quarters load and full load. The inlet
pressures corresponding to the loads are given in Table 6.
The optimal angle of spark firing, θ spark
, and burn duration,
∆θ, for the sets of pressure data are given in Table 7.
The error, approximate area of the actual pressure curve
and the error as a percentage of the area are also given in
the table.
The percent error of the model was approximately 0.5%
for all the trials. Error increases with higher inlet pressures
because of the greater pressures in the data.
Spark and Burn Duration Evaluation. The results of the
spark firing and burn duration determination are shown in
Table 7. Generally, the burn duration angle increases with
load and, with the exception of the quarter load data, the
angle spark firing becomes later with increasing load. The
(9)
delay in spark firing as the inlet pressure increases is
expected, however, the increase in burn duration goes
against the predicted trend.
In the model, an increase in inlet pressure is reflected
as a larger pressure at the spark and a greater mass of fuel
in the cylinder. With more fuel to burn, the model lengthens
the burn time. Burn duration is decreased only slightly by
the increase in pressure at the spark.
In actuality, an increase in the initial pressure during
combustion can greatly alter the way the fuel burns. Higher
pressure in a limited volume means that the temperature in
the cylinder is higher and that the fuel and air molecules are
more reactive. The mixture burns faster, so the burn duration
becomes shorter. The delayed spark firing takes advantage
of the more reactive nature of the fuel to get more
power out of the stroke.
The model does not take into account the increased
temperature at the spark. The reactivity of the fuel is not a
factor in the MLK combustion relation that is used by the
model. Under higher inlet pressure, the gases in the cylinder
are hotter throughout combustion and the combustion,
which is an oxidation reaction, occurs faster. This increase
in reaction rate is due to the increased kinetic energy so it
takes a smaller amount of additional energy to reach the
activation energy for the reaction. 6
The MLK and similar burn fraction models were originally
formulated from burn profile pictures. These relations
were empirically determined by observing the photographs,
which means that the models were not based on principles
of thermodynamics. This fact caused the error in the model.
An improvement would be to find a combustion model based
on the work-energy theorem where pressure and temperature
are related to the total energy of the system.
Summary
This model predicts the pressure of the gas in a cylinder
by incorporating the isentropic gas process model, the
McCuiston, Lavoie and Kauffman model and the gas flow
through an orifice model. It predicts the pressure in the cylinder
fairly accurately. The most significant weakness of the
model is that is does not take the changes in fuel reactivity
during combustion into account. However, the model is
simple and can be used to give quick estimates of pressure
for other models, such as oil film thickness.
Acknowledgments
This project would not have been possible without the
help and advice of Professor John Heywood and Tian Tian
at the Sloan Automotive Laboratory at MIT. Thanks also go
to Bouke Noordzij and Peter Hinze for supplying engine data.
The author is grateful to Mr. Don Hyatt for providing the use
of the Thomas Jefferson High School Computer Systems
lab. Lastly, the author thanks Dr. John Dell for his support
and help through all the difficulties.

145 Journal of Undergraduate Sciences
Engineering Sci.
References
(1) Heywood, J. B. 1988. Internal Combustion Fundamentals (New
York: McGraw-Hill, Inc.).
(2) Gyftopoulos, E. P., and G. P. Beretta. 1991. Thermodynamics:
Foundations and Applications (New York: Macmillan Publishing
Company).
(3) Cheung, H. M., and J. B. Heywood. 1993. “Evaluation of a One-
Zone Burn-Rate Analysis Procedure Using Production SI Engine
Pressure Data.” SAE paper 932749.
(4) Amann, C. 1985. “Cylinder-Pressure Measurement and Its Use
in Engine Research.” SAE paper 852067.
(5) This equation was suggested by Professor John Heywood.
(6) Rossotti, H. 1993. Fire (New York: Oxford University Press).
Appendix A: Renault Engine Specifications
Appendix B: Graphical Comparisons